Gauge symmetry
The exponential function for the coupling energy e3vΦ,
as derived from experiment (1), represents a capturing of the magnetic
condensate wavefunction through the energy degeneracy of the water ice cage
structures such that it becomes real and observable. It also expresses the
scale-invariant and gauge-invariant properties of the system. Conservation of
angular momentum requires the existence of a sink/ source for the associated
changes in inertia (variable effective radius) together with a corresponding
symmetry relation.
Scale-invariance is attributed to the hyperbolic
curvature of Lorentz boosts that impose a conformal symmetry on the embedding
vacuum manifold [22]. Conformal symmetry is
able to describe the tetrahedral, hydrogen bonded, 3-dimensional spatial
geometry of the crystal-fluid under non-extensive volume changes whilst its
variable hyperbolic surface area maps to the gradient energy term of the
Lagrangian (see below). Since a universality class of topologically invariant
critical exponents has been determined for the continuous (second order) phase
transition, the system can be modelled through conformal field theory in
4-spacetime dimensions, ie. it is describable by a renormalizable quantum field
theory in which the non-perturbative conformal bootstrap is irrelevant [23].
Yang-Mills theory is a strongly coupled quantum
field theory [16], ie. a gauge theory in which
the low-energy dynamics are far removed from any classical description [24]. It is represented through the mathematical
structure of Lie groups that provide for intricate topologies. The compact,
simple Lie group
SU(3) describes the strong interaction in QCD, ie. the
binding of quarks and gluons through confinement mechanisms. The mechanical
action of the piston expander can be described by the emergence of a gauge
field
Φ and the critical length exponent
v (as Equation (4)
below). In QCD such gauge fields are collectively known as gluon fields. The
field strength, or curvature
Fμν, has the general
form:
where
Aν provides for
Lorentz invariance and
Aμ is the gauge connection.
The gauge connection depends upon a complex scaling
symmetry that is exact but not directly observable [25].
In the quantum state Ψ → eiθΨ,
which could be interpreted as a potential sink/ source for the ‘hidden’ inertia
of the false vacuum system (although later this is revealed not to be the
case). It also represents the complex order parameter field of the
Ginzburg-Landau superconducting phase transition included in Equation (7)
below.
Experimental results lead to a relativistic
manifestation of length expansion and time contraction arising from false
vacuum behaviour in a thermodynamically constrained condensed matter system.
The local stability conditions maintained through dynamically responsive
inhomogeneities in this soft matter are deemed equivalent to the property of
asymptotic freedom, or antiscreening, which accounts for the mechanism of
colour confinement in particle physics, ie. scale-invariance is effective
across the micro- and macro-scales. In QCD it is the emergence of clouds of
virtual gluons that establish the antiscreening phenomenon [3]. In both mechanisms, increasing kinetic energy
is mirrored by an increasing negative energy potential such that total energy
remains constant.
Whilst the crystal-fluid material displays high
stability in total energy and density, the embedding manifold always remains on
the threshold of instability. Small positive or negative pressure perturbations
produce divergent critical behaviour manifesting as large variations in swept
volume V. However, this is not the specific volume of the
material system (density remaining almost constant) but rather the
non-extensive volume change associated with the condensation of magnetic
charges and simultaneous emergence of a gradient energy term.
The ‘rolling’ critical response initiated by
anisotropy in water ice cage structures facilitates net energy gain for the
duration of external pressure perturbations, either positive or negative, in a
display of self-organized criticality [26].
The angular momentum of the material is transferred to or from the embedding
vacuum manifold through self-organizing behaviour and high energy degeneracy of
the water ice cage structures. However, this brief statement does not provide a
full description and a more detailed hypothesis follows.
Work derived from the piston expander can be
expressed in terms of an electromagnetic pseudo-scalar gauge-invariantly
coupled to the gauge field
Φ and the critical length exponent
v.
The relationship is in agreement with the cosmological inflation model proposed
by Ratra [27]:
where the covariant vector Fμν
and contravariant gradient potential Fμν combine to
produce Lorentz invariance for the pseudo-scalar field when rotated on a
hyperbolic manifold, ie. the electromagnetic field pseudo-scalar enables a
non-additive energy contribution to enter the non-equilibrium system in the
form of hyperbolic curvature.
In the quasi-micro-canonical ensemble [1], the electromagnetic field pseudo-scalar is
involved in the coupling mechanism but contributes no work in itself. It
expresses the Berry curvature of the vacuum manifold whilst hosting the
magnetic exchange pathways that facilitate energy transfers either to or from
the vacuum manifold. The inner-product of the E and B fields remains the same
viewed in all relativistic frames [28] with the
pseudo-scalar field remaining Lorentz invariant such that:
where
c is the speed of light.
Ginzburg-Landau theory states that the free energy
of a superconductor near a phase transition can be expressed in terms of a
complex order parameter field [29]:
Then a complex rendering of the coupling energy
term
e3vΦ maps to a complex wavefunction of the Berry phase:
where the quantity |
Ψ(
r)|
2
reflects the density of superconducting charge carriers; electrons for
Type-II and the magnetic counterpart arising from gauge monopole charges for
dual Type-I [11].
Appendix A provides a summary of the
Ginzburg-Landau theory of superconductors.
In the dual superconductor model of confinement [9,10], the Yang–Mills vacuum is based on the
condensate of a magnetically charged Higgs field. In this situation, the
critical correlation length ξ also represents the coherence length ξ’
of the magnetic monopole condensate [12] which
diverges to encompass total hyperbolic volume V of the system at
the superconducting phase transition [29]. In
this case the monopole condensate ξ’ becomes exceptionally large under
relativistic Lorentz rotation and expands effectively even further due to the
‘rolling’ critical response [1]. ξ’ also
gives the distance over which the dual superconductor can be represented by a
wavefunction.
Since the coherence length
ξ’ and maximum
value for the Ginzburg-Landau parameter
κ for the Type-I dual
superconductor are known [1], the London
penetration depth
λ can be derived (see
Appendix A).
ξ’ and
λ are equal to the inverse Higgs mass
mH
and inverse vector boson mass
mV, respectively [11]. In normal metallic superconductors
λ is
the distance within which an externally applied magnetic field disappears
inside the superconductor. However, for the dual superconductor
λ
represents a distance beyond the developing QCD flux tubes within which the
magnetic current and electric field are expelled as a result of the dual
Meissner effect.
So, mH determines the extent of
QCD vacuum, which manifests in the embedding vacuum manifold volume V
and mV determines the Gaussian hyperbolic curvature K of
the embedding vacuum manifold.
Appendix A
includes supporting quantitative analysis.
The complex form of the coupling energy term
resembles a quantum mechanical wavefunction in which the energy spectrum is
made entirely real and observable through dissipative structuring of water ice
cages. Ψ0(r) corresponds to the emergence of
the gauge field Φ at the Type-II to dual Type-I superconducting phase
transition. Dissipation of either the scalar field or the critical correlation
length ξ would represent a collapse in the wavefunction.
Application of de Moivre’s formula and isomorphic
mapping of the complex field to rotational matrix form gives:
and similarly expressing electromagnetic duality as
rotations in the 2-dimensional real plane:
Then the conjugate transpose of (8) is (9) and VVH
= 1 suggesting that PV work of the piston expander is contingent
upon the decomposition of a Hermitian unitary matrix A into two 2 x 2
non-Hermitian unitary matrices (ie. two complex matrices V and VH
containing both real and imaginary components such that VH ≠ V) [30]. The gauge field and the electromagnetic
pseudo-scalar are thereby coupled through a marginal interaction.
Although this interpretation appears at odds with
the expression for
PV work stated in (4), in fact any 2 x 2
complex symmetric matrix A can be eigendecomposed into a diagonal matrix D
sandwiched between two complex unitary matrices, ie. VDV
H in this
case. Minkowski spacetime vectors can be represented by 2 x 2 orthogonally
diagonalizable matrices and incorporated into the extended physical VDV
H
decomposition to reveal the coupling energy source:
These Hermitian matrices exhibit basic
3-dimensional rotation as well as 4-dimensional Lorentz transformation
properties consistent with the relativistic length expansion and time
contraction associated with the non-extensive element of PV work,
as revealed through the experimental results [1].
Thus, the 2 x 2 unitary matrix A as a member of the U(2) symmetry group
is decomposed into factors identifiable as both Hermitian and non-Hermitian.
When represented in terms of gauge symmetry groups [30], the
U(1) group of electromagnetism (via
its mapping to
SO(2) in the 2-dimensional real plane) and the
SU(2)
group of the complex order parameter
Ψ(
r), are in fact
subgroups of the
U(2) group such that:
which describes a mapping to a Yang-Mills
electroweak symmetry group [31] where ℤ
2
represents the topology associated with the condensation of gauge monopoles [32]. Formation of the
U(2) group is
accompanied by critical behaviour and emergence of the gauge field
Φ as
predicted by the Yang-Mills theory.
The dual superconductor model has several
interpretations that require condensation of gauge monopoles, just as normal
superconductivity results from the condensation of electric charges (or Cooper pairs–see
Appendix A ) [11,12]. Theoretical frameworks for the condensation
of gauge monopoles have been structured in terms of Abelian gauge-invariance
(the
SU(2) gauge symmetry group) or non-Abelian gauge-invariance (the
SU(3)
gauge symmetry group). Recent efforts [12]
have sought to extract the Abelian component responsible for gauge-independent
quark confinement from non-Abelian gauge-invariance required for gauge monopole
condensation without losing the essential characteristic of asymptotic freedom.
From the experimental findings [1] such a
solution emerges out of an electroweak interaction that preserves asymptotic
freedom, as described below.
In the vacuum of a dual superconductor, the dual
Meissner effect compresses the chromoelectric flux between a quark and
antiquark into a thin flux tube to form the hadronic string [11,33]. As the distance between quark and antiquark
increases, the flux tube becomes longer whilst maintaining a minimal thickness.
This geometry ensures that the energy increases linearly with length to create
a linear confining potential between the quark and antiquark that bears a
similarity to the linear oscillating Hamiltonian of the system. The flux tube
determines the extent of QCD vacuum suppression, ie. positions where the
colour-electric field is maximally expelled to leave a residual dual superconductivity
[34].
Yang-Mills theory requires the existence of both
chromomagnetic monopole condensation (given by a coherence length) and the dual
Meissner effect (given by a penetration depth) [11,12].
The force carrying gauge bosons of QCD are gluons which perform a similar role
to photons in electromagnetism. Since the gluon field represents a local
expulsion of the QCD vacuum, the absorption of physical gluon emissions into
the QCD vacuum would tend to reduce local ‘space density’ and effective
magnetic permeability μ0. The net effect is to reduce the
hyperbolic curvature of the embedding manifold, ie. a quantum mechanical
process manifests as ‘strong gravity’ [35].
Berry phase and parity-time (PT) symmetry
The foundations of Berry phase physics lie in the
adiabatic theorem of quantum mechanics [36]
which provides a formal description for a system coupled to a slowly changing
environment. If the system Hamiltonian
H(
t) varies adiabatically
and │
Ψ(
t) ⟩
is an associated eigenstate then, following cyclic evolution of the
environmental parameters where
H(
T) =
H(0), the state
returns to itself but gains an additional phase factor [36,37]:
where
α represents the angular momentum of
the wavefunction. It originates from the exclusion of momentum resulting from
chromomagnetic monopole condensation which is effectively stored in the
electromagnetic field pseudo-scalar [24].
The adiabatic theorem is based upon a single,
non-degenerate eigenstate to which the system ‘clings’ as the environment is
slowly changed [38]. However, for the
pressure-perturbed system being examined, asymptotic freedom constrains
innumerable, degenerate and excited eigenstates to a singular value of total
energy in the oscillating Hamiltonian function. External pressure perturbations
applied to the crystal-fluid material see changes in kinetic/ internal energy
mirrored by changes in negative energy potential such that total energy remains
constant. When perturbations cease, the negative energy potential dissipates
but internal energy becomes fixed close to the final resting value. So, in this
case, a positive or negative perturbation of any duration is responsible for a
single linear oscillation, or cycle, that is imposed upon a linear,
sliding-scale of discrete values. Integration of the scalar potential ∇Φ over a Hamiltonian
cycle reveals the gradient energy term -½( ∇Φ)2 that becomes
observable in the PV work extracted from the piston expander (see
Equation (22) below).
The transient negative energy potential responsible
for the phase factor
Ψ(
T) exists only for the duration of the
pressure perturbation. Whilst the Hamiltonian remains constant under
perturbation (as
Figure 2), it resolves to a different, stable value
once the perturbation ceases. The final value of internal energy is then
‘propped’ and stabilized through dissipative structuring of water ice cages,
subject to limited dielectric relaxation, as quantified by Stage 2-3 and Stage
4-1 (
Figure 2).
For acceleration (Stage 1-2) the effective radius
decreases, and for deceleration (Stage 3-4) the effective radius decreases.
However, the resulting ‘hidden’ inertia is deemed not to be responsible for the
Berry curvature term within the geometrical phase (7) since the
externally-induced momentum manifests entirely in non-additivity of the
hyperbolic curvature. Instead, the Berry curvature is linked to the
condensation of magnetic charges whereby the resulting exclusion of charge
momentum manifests in the energy potential of the electromagnetic field
pseudo-scalar (5). The Berry curvature is subsequently captured to be made real
and observable in the variable hyperbolic volume of the embedding vacuum
manifold. Again, this hyperbolic volume is stabilized by the dissipative
structuring of water ice cages within the crystal-fluid material so that the
complex Berry phase is transformed into real work done.
For a classical thermodynamic system, changes in
inertia ½mr2 represent changes in kinetic/ internal energy.
However, since both internal energy and specific volume are highly constrained
parameters within a false vacuum system, the energy of acceleration/
deceleration is prevented from manifesting in the crystal-fluid material. Thus,
Pv work is limited to interactions with the walls of the vessel. For the
synchronized U(2) symmetry group identified below, angular momentum is
instead conserved in the acceleration/ deceleration of quarks that results in
the emission/ absorption of gluons, ie. changes in negative energy potential.
Gluons emitted by quarks are absorbed by the QCD vacuum manifold whilst the
gluons absorbed by quarks emerge from the QCD manifold, thereby tending to
effect local ‘space density’ and effective permeability μ0.
Pv is insignificant in comparison to
PV such that it represents the negative energy potential of the crystal-fluid
material only. Therefore, for a constant Hamiltonian oscillator of constant
mass
m, ½
r2 ∝
1/
Pv, as described in Appendix B
of [1]. The average 1-dimensional radius
rx
of the stable, non-critical system is then found:
The Gaussian curvature
K for the
2-dimensional, hyperbolic surface of the non-critical system (ie. with no
topological defects) for the principal curvature relationship of
rx
= -
ry, can then be determined:
or
Then, the average Gaussian radius of hyperbolic
curvature (1/
K or
Rg) is given by:
Principal curvature
K has units of m
2 s
-2
that map directly to the vector boson mass
mV as the inverse
of the penetration depth
λ (as described in
Appendix A). Through this mechanism, the
negative energy potential of gluons is conserved through indirect hyperbolic
curvature quantifiable by the non-equilibrium values of pressure
P and
specific volume
v. Thereby, a quantum mechanical action can be tuned
thermodynamically under false vacuum conditions.
Decomposition of the complex gauge connection (e
iΦ)
3ν
in equations (8) and (9) suggests that complex Berry curvature is necessary for
emergence of real coupling energy (1). It also determines the phase of
electromagnetic duality, which in the extreme leads to dual superconducting
behaviour, ie. condensation of magnetic charges resulting in the exclusion of
magnetic current and the electric field
1. The cyclic
evolution of the gauge connection results from the effective adiabatic property
of the constrained false vacuum system (as revealed in the constant Hamiltonian
oscillator of
Figure 2) to establish a novel form of the Berry phase [36], one responsible for topological ordering in
the dual Type-I superconductor [8]. As with
the conventional ground-state Berry phase, this ‘excited-states’ variant
exposes the gauge structure in quantum mechanics [13,39].
In addition to describing the emergence of a gauge
field Φ, the
gradient energy term -½(∇Φ)2 of the
Lagrangian also maps to the complex order parameter field Ψ(r)
in accordance with Ginzburg-Landau theory. The PV work
generated in the piston expander suggests that the associated quantum
mechanical wavefunction is made real and observable, a phenomenon recently
uncovered by Gu et al. [13]. More
precisely, the VDVH decomposition reveals that the wavefunction
becomes entirely real as the coupling energy is exposed through the diagonal
matrix D in the VDVH decomposition.
Since the system can be described through a
combination of Hermitian and non-Hermitian matrices, it resembles a PT
symmetric system [14]. Such systems are
characterized as not being isolated from the environment (ie. non-adiabatic)
but subject to highly constrained interactions. This description is consistent
with the false vacuum behaviour of the crystal-fluid material where both specific
volume and internal energy are highly constrained. Energy and entropy gains and
losses to the environment (including the embedding vacuum manifold in this
case) are exactly balanced, ie. a renormalized, scale-invariant interaction
between condensed matter and quantum wavefunction becomes evident in the
constant energy Hamiltonian.
PT symmetry requires both space reflection and time
reversal symmetries. The upside-down potential of the quartic term as
identified by Bender [14] is consistent with
the marginal interaction and negative gradient energy term derived from
experiment [1]. However, the results presented
here reveal the symmetry of Lorentz boosts, ie. symmetries in the expansion and
contraction of both space and time, which may represent a more generalized form
of PT symmetry.
1 In this case hyperbolic curvature
of the vacuum manifold establishes dual superconductivity rather than the
hydrogen bonding induced curvature of the crystal-fluid lattice initiating the
superconducting phase transition, as suggested in [1], ie. the causality is reversed.
Symmetry synchronization and conserved quantities
Quark acceleration produces gluon emissions since a
lower binding potential is necessary to maintain the momentum and energy of any
given quark colour configuration [40]. This
results from a gluon recombination process whereby a quark and antiquark pair
are annihilated. The emergence and absorption of physical gluons represents an
exchange between the non-Abelian gauge symmetry of QCD and the Abelian gauge
symmetry of the vacuum manifold, ie. an electroweak interaction.
The following non-Abelian Faddeev-Niemi
decomposition is considered [41]:
This decomposition is a restricted one since
splitting and recombining gluons in
SU(3) represents a limited
interaction with a
U(2) spacetime manifold rather than full symmetry
breaking to
SU(2). The requirement for a Higgs-type scalar field is
satisfied by the emergent gauge field
Φ [12].
So:
Asymptotic freedom is thereby maintained through
the dominant SU(3) group. Again, U(2) appears as an electroweak
symmetry group [31] with ℤ2
representing a topology consistent with the condensation of gauge monopoles [32].
A U(2) gauge symmetry that provides for the
condensation of gauge monopoles has so far been identified in both the
condensed matter system and the underlying QCD particle physics. However, it is
also possible to determine a U(2) gauge symmetry for the vacuum manifold
of local spacetime through which hyperbolic curvature and scalar field
potential are effected. That is, where the splitting and recombining of
force-carrying gluons are associated with fictitious forces in non-inertial
reference frames.
The Lorentz group
SO(4) provides for the
conservation of energy and angular momentum in 4-dimensions (ℝ
4) through
two continuous symmetries; rotations in 3-dimensional Euclidean space and
Lorentz boosts which influence both space and time [42].
The 4 x 4 orthogonal matrix representation of the metric tensor can also be
cast in terms of a 2 x 2 unitary matrix operating on a complex 2-component
spinor. The complete unitary 2 x 2 transformation matrix for spinor rotations
and boosts can be expressed as:
or
where
θ is the Lorentz rotation angle,
σ is the Pauli spin matrix, and ϕ
is
the angle associated with the Lorentz boost (or rapidity) [43].
Equation (19) represents a ‘right-handed’ spinor
ϕR
and (20) represents a ‘left-handed’ spinor
ϕL, ie. the Weyl
spinors. Later insights by Dirac led to the concept of the bispinor which,
unlike (19) and (20), preserves parity of the wavefunction under the sign
reversal operation
Ψ(
x,
t) →
Ψ(-
x,
t)
thereby maintaining a positive gauge field and positive energy (whilst also
predicting the existence of antimatter). However, retaining the 2 x 2 unitary
matrix whilst acknowledging parity preservation requirements produces the
following spacetime group representation [44]:
The symmetry group decompositions in (11), (18) and
(21) are then amalgamated to describe a consolidated ‘symmetry synchronization’
that establishes common scale- and gauge-invariance in
U(2), as shown
schematically in
Figure 4:
When a symmetry is broken, a corresponding order
parameter that diminishes to zero can often be identified. However, in this
case the complex order parameter Ψ(r) emerges where
symmetry is synchronized.
Both energy and angular momentum are conserved
within the common U(2) group to reveal the time and space symmetries of
a Lorentz boost in agreement with Noether’s theorem (see below). Since there is
a gluon field for each colour charge, it follows that each gluon field can be
composed of a time-like component and three space-like components. These
components relate to the electric potential and the magnetic potential,
respectively, and will interact with the vacuum manifold to determine the
values of effective permittivity ε0 and effective
permeability μ0.
Variations in effective μ0
require that a spontaneous magnetic flux Ms, with associated
spontaneous magnetic field Hs, emerges to conserve magnetic charge.
Fractionalized magnetic charges arising from the geometrically frustrated
crystal-fluid material can be interpreted as condensing into a gauge monopole
topology that excludes magnetic current to provide magnetic exchange pathways.
The correlation length ξ of the magnetic monopole condensate produces
divergent critical behaviour that is shown to have a distinctive universality
class of critical exponents. The gauge monopole topological defects act as both
convergent sinks (under acceleration) and divergent sources (under
deceleration) of the magnetic flux Ms [8].
The nature of these defects is speculated in Appendix
A.
Similar principles apply to variations in vacuum
energy determined by the local ‘space density’ (which determines the embedding
manifold curvature). Conservation of energy requires that negative
PV
work is performed under false vacuum acceleration (energy is transferred to the
vacuum manifold) whilst positive
PV work is performed under
deceleration (energy is transferred from the vacuum manifold). Work is related
to the gauge/ scalar field
Φ as follows [1]:
The right-side of Equation (22) represents the
gradient energy term of the Lagrangian function resulting from the scalar
potential ∇
Φ
developed across the gauge monopole topology to give the integral of the scalar
potential ∇
Φ. The
Lagrangian action of the left-side, ie. mechanical work, is also related to the
critical response function revealed in the coupling relationship (1) to confirm
renormalization in the synchronized
U(2) group complex parameter field
Ψ(
r).
That is, energy equivalence between the long-range dissipative structuring of
water ice cages and the short-range confinement mechanisms of sub-atomic
particles, as illustrated in
Figure 4. This outcome aligns with
Anderson’s speculative prediction [45]:
‘Physics in the 20th century solved
the problems of constructing hierarchical levels which obeyed clear-cut
generalizations within themselves […]. In the 21st century one revolution which
can take place is the construction of generalizations which jump and jumble the
hierarchies, or generalizations which allow scale-free or scale transcending
phenomena. The paradigm for the first is broken symmetry, for the second
self-organized criticality.’
With
U(2) scale- and gauge-invariance
spanning the asymptotically-free behaviour of both the macro-scale dual
superconducting system and the quark-gluon system via interactions with the
embedding vacuum manifold, a physical correspondence between non-equilibrium
thermodynamics and quantum mechanics is established. Since the superconducting
phase transition is represented by Ginzburg-Landau theory (ie. gauge-invariant
coupling of a scalar field to the Yang-Mills action is predicted) it seems
reasonable to link the gradient energy gap of
Figure 3 to the mass gap
problem in QCD.
Gapped and gapless topologies
The results in
Figure 3 show emergence of
the gauge field
Φ
as a gap between Type-II superconductivity on the left and dual Type-I
superconductivity on the right. This represents a transition between the gapped
state of the magnetically ordered Type-II superconductor and gapless state of
the topologically ordered dual Type-I superconductor. At this point, the gauge
monopole charges condense and the electric field is excluded to be confined on
the surface of the system, ie. prior to the emergence of topological defects
that penetrate the magnetic condensate. That is, a gapless surface is
established so that the Berry phase manifests as a non-trivial topological
insulator [36].
The gapless surface may be protected from external
perturbations tending to re-open the gap through non-Abelian topology, as
represented by the ℤ2
Chern number in the symmetry group decompositions of (11) and (18). In a review
of topological superconductors [46], Sato and
Ando explore the connection between ℤ2
and time reversal symmetry that is consistent with the symmetry of Lorentz
boosts described above. The ℤ2
Chern number can be interpreted as a U(2) group that fibres over a
circle as a 3-sphere bundle, ie. a Hopf fibration results [47].
Typically, a topological insulator is characterized
by a non-robust, non-degenerate ground-state in which energy bands coincide and
exceptional, or ‘diabolical’, points occur. However, the Berry phase variant
identified above displays the following features: asymptotic behaviour
(robustness against perturbations); a critical correlation length ξ
(long-range entanglement); conformal geometry (describable through quantum
field theory); and degeneracy in non-trivial topology on a hyperbolic manifold [47]. Thus, the system also appears to be
topologically ordered and so describable by an effective, low-energy
topological quantum field theory (TQFT) in which many-body states have
topological ground-state degeneracy [48]. In
TQFT the critical correlation length ξ is topologically invariant and
therefore insensitive to the geometry of the embedding manifold, ie. the
critical exponents within the universality class remain constant under Lorentz
boosts.
Within the research field of topological phases of
matter, as investigated to date, all the topologically ordered states realized
experimentally or investigated theoretically are established through strong
electron-electron interactions. The coinciding valence bands of gapless
‘diabolical’ points allow for degenerate electron movements between the bands.
In a crystal structure, the electronic band structures are described by Bloch’s
theorem as expressed by:
where
Ψ is the wavefunction,
r
is position,
u is a periodic function, and
k is the crystal
momentum vector.
However, the original formulation of the Berry
phase was not specifically related to Bloch electrons. Instead, it was based on
the general idea that quantum adiabatic transport of particles in slowly
varying fields (eg. electric, magnetic, or strain) could in principle modify
the wavefunction by terms other than just the dynamical phase. So, Equation (23)
is seen to map to the experimentally derived Equation (7) where variable
electromagnetic duality of the pseudo-scalar field gives the periodic function u(r)
and (eiΦ(r))3v represents the angular
momentum of magnetic charges that become effectively stored in the
pseudo-scalar field as the magnetic monopole condensate forms [24].
Since the degeneracy associated with the
crystal-fluid material occurs in metastable excited-states (with non-zero
temperatures), the gapless degeneracy of the crystal-fluid material cannot be
attributed to Bloch electrons. So, whilst the gapless surface of the dual
superconductor is protected through ℤ2
topology, an additional mechanism is necessary to supress excited-state
fluctuations such that Bloch-wave behaviour can emerge.
A potential solution is presented in
Figure 4
where the synchronized continuous symmetry group
U(2) leads to
descriptions of asymptotic freedom in both quantum and condensed matter
systems. As conservation of angular momentum extends into the microscopic
quantum realm, so confinement mechanisms extend out into the macroscopic
condensed matter of the dual superconductor under a renormalized Noether
symmetry. The non-Abelian
SU(3) group of QCD remains dominant so that
excited-state fluctuations due to changes in momentum are suppressed through
confinement mechanisms and the constant Hamiltonian function is preserved.
Topological defects may represent the penetration of the dual superconductor by
the excluded electric field in the form chromoelectric flux tubes, so enabling
the formation of quark-antiquark pairs together with an inherent confinement
mechanism.
Acceleration and deceleration of the crystal-fluid
material thereby become confined interactions responsible for the splitting and
recombining of gluons. Gluons are either absorbed by or emerge from the QCD
vacuum manifold. The Higgs-like gauge field
Φ
also emerges in the transition from gapped Type-II superconductivity to gapless
dual Type-I superconductivity to establish a reciprocal gap in the gradient
energy, as revealed in
Figure 3. For the gapless state, the spin-1
vector gluons are the force-carrying
SU(3) gauge bosons/ quasiparticles
that generate the Bloch-wave description [49].
Gluon interactions with the embedding QCD vacuum manifold enable fictitious
forces to emerge in non-inertial reference frames that lead to
PV
work in the piston expander.
Whilst the massless gluons either emitted or
absorbed by accelerating or decelerating quarks act as gauge particles, gauge
invariance is only established where the associated gauge field can emerge
under U(2) ‘symmetry synchronization’ at a critical correlation length ξ
in the low-energy system. That is, gauge monopole charges are required to
condense such that the scalar field potential induces a flow of magnetic
charge.
By extension, the lower bound of the QCD mass gap
can be attributed to a symmetry-breaking of U(2) and the evaporation of
a gauge monopole condensate from below in the low-energy, infrared limit. In
this event, the conservation pathway between the condensed matter and quantum
systems no longer exists since the systems are effectively isolated except for
weak residual gravitational interactions, ie. the systems are decoupled. With
this spontaneous breaking of U(2) gauge symmetry into isolated
sub-groups, and the collapse of any critical magnetic correlation length ξ,
there is no mechanism through which non-equilibrium angular momentum can be
conserved through gluon interactions and the corresponding curvature imposed on
the embedding manifold, ie. conservation of energy and momentum becomes limited
to the individual symmetry groups and the condensed matter system becomes
describable by classical thermodynamics and discontinuous (first order) phase
transitions. Thus, in this interpretation of broken gauge symmetry, a strong
gravitational interaction [35] is replaced by
a far weaker one essentially limited to gluon interactions arising from quantum
fluctuations and other irrelevant interactions.
Cosmological analogy
The same principle may be applied to the
high-energy bound of the QCD mass gap from above. Guth’s model of cosmological
inflation [50,51] is also founded upon false vacuum and negative
pressure conditions. The associated deceleration of inflationary expansion
within a false vacuum would result in gluon splitting where conservation of
energy and momentum are mediated by a synchronized order parameter field/
symmetry group together with the corresponding cosmic monopole condensate
necessary to facilitate a critical coherence length and penetration depth. The
splitting of gluons and the emergence of quark-antiquark pairs act to increase
the strong interaction through confinement mechanisms.
The non-extensive inflationary volume expansion
model is [1]:
where
Vr is the reduced volume │(
V
–
Vc)/
Vc│ at the critical temperature
Tc
with a correlation length exponent
v in 3-dimensions. Critical volume
Vc
and critical temperature
Tc are not absolute values but
rather ‘rolling’ dynamical values determined by structural anisotropy and
dissipative false vacuum restructuring of elementary particles under
non-equilibrium conditions.
Vr can also be determined by the
Higgs mass
mH which is the inverse of the coherence length
ξ’.
At the collapse of critical behaviour,
Vr = 1 and
non-extensive inflationary volume expansion ceases.
The critical length exponent
v represents the
dimensionless group parameter of rapidity such that its emergence on a Lorentz
hyperbolic manifold (ie. a Lorentz boost) is associated with a relativistic
length expansion to describe non-extensive inflationary volume expansion [1]. Increasing the effective radius of structural
elements under false vacuum conditions produces deceleration such that angular
momentum is removed from the sub-atomic quarks. The counteracting emergence of
quark- antiquark pairs
qq plus gluons from the QCD vacuum, the associated gluon splitting, together with
subsequent integration of the emergent particles into complex binding
arrangements, establishes the colour and quark confinement mechanisms. These
tend to increase local ‘space density’ and effective magnetic permeability
μ0.
The resultant effect sees hyperbolic curvature of the embedding vacuum manifold
increase leading to inflationary expansion. The process is shown as an analogue
of the experimental findings in
Figure 5.
At the collapse of critical behaviour, the
synchronized gauge symmetry is broken, the cosmic monopole condensate with
associated topology evaporates, and critical correlation length is destroyed.
Absorption of the gauge field into the QCD colour field attributes mass to the
emergent quark and antiquark pairs qq
of the confinement process under a Higgs-like mechanism associated with an
electroweak interaction, whilst volume and internal energy are ‘propped’ and
stabilized by reorganizing dissipative structural elements. Again, under such
an interpretation, classical thermodynamics essentially separates from quantum
mechanics to leave only weak residual gravitational interactions, ie. the
strong interaction becomes short-ranged. The energy and mass of the strong
interaction are thereby effectively fixed at the point where the common
symmetry group associated with cosmological inflation is broken, the cosmic
monopole condensate evaporates, and the critical correlation length is
destroyed.